mathematics

G.H. Hardy is one of the very few mathematicians who’s been immortalized in song, by the Embarrassment no less:

Hardy’s little book, A Mathematician’s Apology (that’s apology in the sense of defense, not regret), written in 1940 near the end of his career, is an eloquent and concise statement of the mathematician’s, theoretician’s, and Platonist’s worldview. It is worth reading especially by anyone who is not a member of those clubs. It is the memoir of a person who has spent so much time discovering theorems of numbers, formulae, and equations that they have come to seem far more real than the discovery of a new species of plant or a new planet, which after all is just one more instance of a form that was already known.

I am not exaggerating on the Platonist front. Hardy states it plainly:

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. A reader who does not the philosophy can alter the language: it will make very little difference to my conclusions.

This is the key point, never to be forgotten. This mathematical reality is more real to him than the world we appear to inhabit. Hardy witnessed connection to this reality in an even stronger form in his friend Ramanujan, the great mystic mathematician, who had sent him a sample his unpolished but noetically brilliant work. Two other mathematicians had dismissed Ramanujan’s work, but on seeing the unknown Ramanujan’s work, Hardy recognized him for what he was and brought him to Cambridge. I have no doubt that Ramanujan cemented Hardy’s Platonism: Hardy rated Ramanujan as the most talented mathematician he had ever known.

Correspondingly, the prose is a mixture of plainspoken simplicity and blatant elitism, cosmic humility and human arrogance, as though Moses had come down from the mountain without desiring to convince anyone that he was right…or not even being sure that he could. His very opening suggests he has only come down from the mountain because his powers have faded and failed him:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

Yet in the human world, mathematics seems a talent of marginal utility (at least to Hardy), and he defends his mortal life by saying only that he did mathematics because he was good at it.

Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created is undeniable: the question is about its value.

It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

As W. J. Turner has said so truly, it is only the ‘highbrows’ (in the unpleasant sense) who do not admire the ‘real swells’.

But in terms of the greater pageant of time, the mathematician has the greatest chance at immortality. He doesn’t compare his field to the empirical sciences (though he looks down on applied mathematics), but I gather that he is more confident of mathematical achievements because their results cannot be overturned by things like as-yet-undiscovered evidence. As for language and literature, they are merely human creations and even more evanescent.

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of satisfying them than a mathematician. His subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Could lines be better, and could ideas be at once more trite and more false? The poverty of the ideas seems hardly to affect the beauty of the verbal pattern. A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.

Indeed, the entire world of the contingent, the observed, the evidentiary, seems instilled with a frailness that makes it ephemeral and far less meaningful. Any connection to the everyday nominal world is something that endangers the solid rock of eternal truths which Descartes described as the sole object of posthumous contemplation. (Our memories do not exist after death, for Descartes, so the only things our souls can contemplate are a priori truths: mathematical and logical ones.)

It is quite common, for example, for an astronomer or a physicist to claim that he has found a ‘mathematical proof’ that the physical universe must behave in a particular way. All such claim, if interpreted literally, are strictly nonsense. It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics.

We can describe, sometimes fairly accurately, sometimes very roughly, the relations which hold between some of its constituents, and compare them with the exact relations holding between constituents of some system of pure geometry. We may be able to trace a certain resemblance between the two sets of relations, and then the pure geometry will become interesting to physicists; it will give us, to that extent, a map which ‘fits the facts’ of the physical world. The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptations, he will be abandoning his purely mathematical position.

And so applied mathematics is inferior to pure mathematics because it is hamstrung by contingent particulars. Airborne truth is brought down to earth by the accumulated weight of midges and gnats:

One rather curious conclusion emerges, that pure mathematics is one the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot.

But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

“Fancy” and “facts” being somewhat self-effacing language, since by this point it is clear that for Hardy, fancy is more enduring than fact. And for anyone who works in these fields long enough, it is hard to imagine how a mathematician could not end up a Platonist after working so dutifully with non-material, abstract entities that constantly produce new, surprising, emergent properties.

This is not a new attitude; the Pythagorean cult is only one of the oldest known manifestations of this tendency. And it exists today in hardly a different form: the “quant” of finance describes being sucked into the world of mathematical reality in a similar though less eloquent way. And the insistence with which string theorists proclaim that their equations are so perfect that they simply must describe the ultimate truth of reality is more or less just a variation on Hardy’s ideas of theoretical elegance and beauty.

C.P. Snow knew Hardy and Hardy thanks Snow in the book, but the book belies Snow’s famous generalization about the two cultures of humanities and science. To hear Hardy tell it, the real divide is not between the humanities and the sciences but between the theoreticians and the engineers, idea and praxis, rationalists and empiricists, philosophers and storytellers, gnostics and skeptics.

It is more a continuum than it is a dichotomy, but each pole is a strong attractor and tends to draw in those who already lean toward it. As someone who by temperament or talents has always tended to fall closer to the engineer’s side, I always hope for the theorists to remember that suffering is as real as any theorem. Hardy refers to the anodyne of escape provided by theory, but not only can it also be a dereliction of human duty, but it is also ultimately an unreliable respite for mere particulars such as ourselves:

There is one purpose at any rate which the real mathematics may serve in war. When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, “one at least of our nobler impulses can best escape from the dreary exile of the actual world.” It is a pity that it should be necessary to make one very serious reservation—he must not be too old. Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.